A338692 Positions of 1's in A209615.

1, 4, 5, 6, 9, 13, 14, 16, 17, 20, 21, 22, 24, 25, 29, 30, 33, 36, 37, 38, 41, 45, 46, 49, 52, 53, 54, 56, 57, 61, 62, 64, 65, 68, 69, 70, 73, 77, 78, 80, 81, 84, 85, 86, 88, 89, 93, 94, 96, 97, 100, 101, 102, 105, 109, 110, 113, 116, 117, 118, 120, 121, 125, 126

Offset: 1

Author

Jianing Song, Apr 24 2021

Keywords

• nonn
• easy

Comments

Also positions of 1's and 4's in A003324.

Also positions of 0's in A292077. - Jianing Song, Nov 27 2021

Numbers of the form (2*k+1) * 2^e where k+e is even. In other words, union of {(4*m+1) * 2^(2t)} and {(4*m+3) * 2^(2t+1)}, where m >= 0, t >= 0.

Numbers whose quaternary (base-4) expansion ends in 100...00 or 1200..00 or 3200..00. Trailing 0's are not necessary.

There are precisely 2^(N-1) terms <= 2^N for every N >= 1.

Equals A016813 Union A343501.

Complement of A338691. - Jianing Song, Apr 26 2021

Examples

			14 is a term since it is in the family {(4*m+3) * 2^(2t+1)} with m = 1, t = 0.
16 is a term since it is in the family {(4*m+1) * 2^(2t)} with m = 0, t = 2.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..8192
• Kevin Ryde, Iterations of the Alternate Paperfolding Curve, see index "TurnLeft" with a(n) = TurnLeft(n-1).

Crossrefs

Cf. A209615, A338691 (positions of (-1)'s), A016813 (the odd terms), A343501 (the even terms), A003324, A292077, A343500.

Programs

Mathematica

A338692Q[k_] := JacobiSymbol[-1, k]*(-1)^IntegerExponent[k, 2] == 1;
Select[Range[200], A338692Q] (* Paolo Xausa, Feb 26 2025 *)

PARI

isA338692(n) = my(e=valuation(n, 2), k=bittest(n, e+1)); !((k+e)%2)

Python

def A338692(n):
    def f(x): return n+sum(((x>>i)-1>>2)+1 for i in range(1,x.bit_length(),2))+sum(((x>>i)-3>>2)+1 for i in range(0,x.bit_length(),2))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 27 2025

Formula

a(n) = A343500(n)/2. - Jianing Song, Apr 26 2021